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Unformatted text preview: 14.03 Fall 2000 Problem Set 4 Solutions 1. Nicholson 8.5 a. EU = 0.75ln(10000) + 0.25ln(9000) » 9.184 b. By purchasing full insurance at a premium of 250, Ms. Fogg’s wealth in each state becomes 9750. Hence her expected utility is ln(9750) » 9.185, which is greater than her expected utility if she does not buy insurance. c. The maximum amount that Ms. Fogg is willing to pay for full insurance is P s.t. ln(10000 – P ) = 0.75ln(10000) + 0.25ln(9000) By the properties of logs, ln(10000 – P ) = ln ( 10000 0.75 9000 0.25 ) P = 10000  ( 10000 0.75 9000 0.25 )» 259.96 2. Part 1: Both of the risky choices (B and D) have higher expected values than the certain choices (A and C). If Bill were risk neutral or risk loving, he would prefer B to A and D to C. The fact that he is indifferent between them implies that he is risk averse. Part 2: The expected utility of F is EU ( F ) = .25 u (400) + .25 u (900) + .25 u (800) + .25 u (1500) EU ( F ) = .5(.5 u (400) + .5 u (900)) + .5(.5 u (800) + .5 u (1500)) EU ( F ) = .5 EU ( D ) + .5 EU ( B ) = .5 EU ( C ) + .5 EU ( A ) Then note that a 50/50 gamble over C and A has expected value $750. Since Bill is risk averse he will prefer $750 with certainty to this gamble. Hence he prefers E to F. 3. If choices are consistent with expected utility maximization, then there exists some utility function u (.) such that the lottery that gives wealth ( w 1 ,..., w n ) with probabilities ( p 1 ,..., p n ) is preferred to the lottery ( w 1 ',..., w n '), ( p 1 ',..., p n ') iff n n i = 1 p i u ( w i ) > i = 1 p i ' u ( w i ') . Thus A preferred to B implies that u (1m)>.1 u (5m)+.89 u (1m)+.01 u (0), or .11 u (1m)>.1 u (5m)+.01 u (0). And C preferred to D implies that .1 u (5m)+.9 u (0)>.11 u (1)+.89 u (0), or .1 u (5m)+.01 u (0)>.11 u (1m), which is a contradiction. Hence choosing A over B and C over D is inconsistent with expected utility maximization. U ( W ) = 100 W 0.9 4. U '( W ) = 90 W  0.1 U ''( W ) =  9 W  1.1 9 W  0.1 1 a) rr ( W ) = 90 W  0.1 = 10 b) willingness to pay for full insurance is WTP such that U (1  WTP ) = 0.9 U (1) + 0.1 U (0.9) 100(1  WTP ) 0.9 = 0.9(100)(1) 0.9 + 0.1(100)(0.9) 0.9 1 WTP = 1  (0.9 + 0.1(0.9) 0.9 ) 0.9 = 0.01004687 or about $10.05....
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 Spring '10
 çAKMAK
 Microeconomics, Utility

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